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Derivative of:
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Integral of d{x}:
2*x/(1+x^2)
Graphing y =:
2*x/(1+x^2)
Limit of the function:
2*x/(1+x^2)
Identical expressions
two *x/(one +x^ two)
2 multiply by x divide by (1 plus x squared )
two multiply by x divide by (one plus x to the power of two)
2*x/(1+x2)
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2x/(1+x^2)
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2x/1+x2
2x/1+x^2
2*x divide by (1+x^2)
Similar expressions
е^(-2x)/(1+x^2)^(1/2)
-(1+2*x)/(1+x^2)
2*x/(1-x^2)

The derivative of the function/ 2*x/(1+x^2)

Derivative of 2*x/(1+x^2)

The solution

You have entered [src]

 2*x  
------
     2
1 + x

$$\frac{2 x}{x^{2} + 1}$$

(2*x)/(1 + x^2)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = x^{2} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{2 - 2 x^{2}}{\left(x^{2} + 1\right)^{2}}$
Now simplify:

$\frac{2 \left(1 - x^{2}\right)}{\left(x^{2} + 1\right)^{2}}$

The answer is:

$\frac{2 \left(1 - x^{2}\right)}{\left(x^{2} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2  
  2         4*x   
------ - ---------
     2           2
1 + x    /     2\ 
         \1 + x /

$$- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x^{2} + 1}$$

Simplify

The second derivative [src]

    /         2 \
    |      4*x  |
4*x*|-3 + ------|
    |          2|
    \     1 + x /
-----------------
            2    
    /     2\     
    \1 + x /

$$\frac{4 x \left(\frac{4 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /                   /         2 \\
   |                 2 |      2*x  ||
   |              4*x *|-1 + ------||
   |         2         |          2||
   |      4*x          \     1 + x /|
12*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      2              
              /     2\               
              \1 + x /

$$\frac{12 \left(- \frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + \frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of 2*x/(1+x^2)

The graph:

Piecewise:

The solution